Question: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}5x-3y &= 1 \\ 7x-y &= 1\end{align*}$
Solution: Begin by moving the $x$ -term in the second equation to the right side of the equation. $-y = -7x+1$ Divide both sides by $-1$ to isolate $y$ $y = {7x - 1}$ Substitute this expression for $y$ in the first equation. $5x-3({7x - 1}) = 1$ $5x - 21x + 3 = 1$ Simplify by combining terms, then solve for $x$ $-16x + 3 = 1$ $-16x = -2$ $x = \dfrac{1}{8}$ Substitute $\dfrac{1}{8}$ for $x$ back into the top equation. $5( \dfrac{1}{8})-3y = 1$ $\dfrac{5}{8}-3y = 1$ $-3y = \dfrac{3}{8}$ $y = -\dfrac{1}{8}$ The solution is $\enspace x = \dfrac{1}{8}, \enspace y = -\dfrac{1}{8}$.